Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 83 | 87 | theorem ascPochhammer_map (f : S →+* T) (n : ℕ) :
(ascPochhammer S n).map f = ascPochhammer T n := by |
induction' n with n ih
· simp
· simp [ih, ascPochhammer_succ_left, map_comp]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 1,230 | 1,240 | theorem coord_unitsSMul (e : Basis ι R₂ M) (w : ι → R₂ˣ) (i : ι) :
(unitsSMul e w).coord i = (w i)⁻¹ • e.coord i := by |
classical
apply e.ext
intro j
trans ((unitsSMul e w).coord i) ((w j)⁻¹ • (unitsSMul e w) j)
· congr
simp [Basis.unitsSMul, ← mul_smul]
simp only [Basis.coord_apply, LinearMap.smul_apply, Basis.repr_self, Units.smul_def,
map_smul, Finsupp.single_apply]
split_ifs with h <;> simp [h]... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 2,096 | 2,096 | theorem iInf_singleton (a : α) (s : α → β) : ⨅ x ∈ ({a} : Finset α), s x = s a := by | simp
|
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49d... | Mathlib/MeasureTheory/Integral/Layercake.lean | 524 | 543 | theorem Integrable.integral_eq_integral_meas_lt
(f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
∫ ω, f ω ∂μ = ∫ t in Set.Ioi 0, ENNReal.toReal (μ {a : α | t < f a}) := by |
have key := lintegral_eq_lintegral_meas_lt μ f_nn f_intble.aemeasurable
have lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞ := Integrable.lintegral_lt_top f_intble
have rhs_finite : ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t < f a} < ∞ := by simp only [← key, lhs_finite]
have rhs_integrand_finite : ∀ (t : ℝ), t >... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 1,471 | 1,473 | theorem map_uniformity_set_coe {s : Set α} [UniformSpace α] :
map (Prod.map (↑) (↑)) (𝓤 s) = 𝓤 α ⊓ 𝓟 (s ×ˢ s) := by |
rw [uniformity_setCoe, map_comap, range_prod_map, Subtype.range_val]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Order.Lattice
#align_import order.min_max from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# `max` and `min`
This ... | Mathlib/Order/MinMax.lean | 271 | 272 | theorem Antitone.map_max (hf : Antitone f) : f (max a b) = min (f a) (f b) := by |
rcases le_total a b with h | h <;> simp [h, hf h]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.pfunctor.multivariate.W from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e3... | Mathlib/Data/PFunctor/Multivariate/W.lean | 310 | 311 | theorem wDest'_wMk' {α : TypeVec n} (x : P (α.append1 (P.W α))) : P.wDest' (P.wMk' x) = x := by |
cases' x with a f; rw [wMk', wDest'_wMk, split_dropFun_lastFun]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
/-!
# N-ary images of s... | Mathlib/Data/Set/NAry.lean | 103 | 104 | theorem image2_union_left : image2 f (s ∪ s') t = image2 f s t ∪ image2 f s' t := by |
simp_rw [← image_prod, union_prod, image_union]
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Aut
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
#align_import algebra.quandle from "lean... | Mathlib/Algebra/Quandle.lean | 239 | 241 | theorem self_distrib_inv {x y z : R} : x ◃⁻¹ y ◃⁻¹ z = (x ◃⁻¹ y) ◃⁻¹ x ◃⁻¹ z := by |
rw [← left_cancel (x ◃⁻¹ y), right_inv, ← left_cancel x, right_inv, self_distrib]
repeat' rw [right_inv]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Ta... | Mathlib/RingTheory/Valuation/Basic.lean | 569 | 577 | theorem map_add_supp (a : R) {s : R} (h : s ∈ supp v) : v (a + s) = v a := by |
have aux : ∀ a s, v s = 0 → v (a + s) ≤ v a := by
intro a' s' h'
refine le_trans (v.map_add a' s') (max_le le_rfl ?_)
simp [h']
apply le_antisymm (aux a s h)
calc
v a = v (a + s + -s) := by simp
_ ≤ v (a + s) := aux (a + s) (-s) (by rwa [← Ideal.neg_mem_iff] at h)
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.... | Mathlib/MeasureTheory/Group/Measure.lean | 306 | 307 | theorem measurePreserving_div_right (μ : Measure G) [IsMulRightInvariant μ] (g : G) :
MeasurePreserving (· / g) μ μ := by | simp_rw [div_eq_mul_inv, measurePreserving_mul_right μ g⁻¹]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure fr... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 69 | 76 | theorem withDensityᵥ_neg : μ.withDensityᵥ (-f) = -μ.withDensityᵥ f := by |
by_cases hf : Integrable f μ
· ext1 i hi
rw [VectorMeasure.neg_apply, withDensityᵥ_apply hf hi, ← integral_neg,
withDensityᵥ_apply hf.neg hi]
rfl
· rw [withDensityᵥ, withDensityᵥ, dif_neg hf, dif_neg, neg_zero]
rwa [integrable_neg_iff]
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "... | Mathlib/NumberTheory/NumberField/Norm.lean | 72 | 85 | theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x) ↔ IsUnit x := by |
classical
refine ⟨fun hx => ?_, IsUnit.map _⟩
replace hx : IsUnit (algebraMap (𝓞 K) (𝓞 L) <| norm K x) := hx.map (algebraMap (𝓞 K) <| 𝓞 L)
refine @isUnit_of_mul_isUnit_right (𝓞 L) _
⟨(univ \ {AlgEquiv.refl}).prod fun σ : L ≃ₐ[K] L => σ x,
prod_mem fun σ _ => x.2.map (σ : L →+* L).toIntAlgHom⟩ _ ... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.... | Mathlib/Logic/Basic.lean | 857 | 860 | theorem exists₂_comm
{ι₁ ι₂ : Sort*} {κ₁ : ι₁ → Sort*} {κ₂ : ι₂ → Sort*} {p : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Prop} :
(∃ i₁ j₁ i₂ j₂, p i₁ j₁ i₂ j₂) ↔ ∃ i₂ j₂ i₁ j₁, p i₁ j₁ i₂ j₂ := by |
simp only [@exists_comm (κ₁ _), @exists_comm ι₁]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Gro... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 568 | 578 | theorem limsup_eq_tendsto_sum_indicator_atTop (R : Type*) [StrictOrderedSemiring R] [Archimedean R]
(s : ℕ → Set α) : limsup s atTop = { ω | Tendsto
(fun n ↦ ∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → R) ω) atTop atTop } := by |
rw [limsup_eq_tendsto_sum_indicator_nat_atTop s]
ext ω
simp only [Set.mem_setOf_eq]
rw [(_ : (fun n ↦ ∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → R) ω) = fun n ↦
↑(∑ k ∈ Finset.range n, (s (k + 1)).indicator (1 : α → ℕ) ω))]
· exact tendsto_natCast_atTop_iff.symm
· ext n
simp only [Set.ind... |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 142 | 155 | theorem addHaar_eq_zero_of_disjoint_translates_aux {E : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E)
[IsAddHaarMeasure μ] {s : Set E} (u : ℕ → E) (sb : IsBounded s) (hu : IsBounded (range u))
(hs : Pairwise (Disjoint on fun n => {... |
by_contra h
apply lt_irrefl ∞
calc
∞ = ∑' _ : ℕ, μ s := (ENNReal.tsum_const_eq_top_of_ne_zero h).symm
_ = ∑' n : ℕ, μ ({u n} + s) := by
congr 1; ext1 n; simp only [image_add_left, measure_preimage_add, singleton_add]
_ = μ (⋃ n, {u n} + s) := Eq.symm <| measure_iUnion hs fun n => by
simpa... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpa... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 207 | 207 | theorem log_nonpos_iff (hx : 0 < x) : log x ≤ 0 ↔ x ≤ 1 := by | rw [← not_lt, log_pos_iff hx, not_lt]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 115 | 119 | theorem hasDerivAt_sqrt_mul_log {x : ℝ} (hx : x ≠ 0) :
HasDerivAt (fun x => √x * log x) ((2 + log x) / (2 * √x)) x := by |
convert (hasDerivAt_sqrt hx).mul (hasDerivAt_log hx) using 1
rw [add_div, div_mul_cancel_left₀ two_ne_zero, ← div_eq_mul_inv, sqrt_div_self', add_comm,
one_div, one_div, ← div_eq_inv_mul]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 766 | 775 | theorem tendstoLocallyUniformlyOn_iff_filter :
TendstoLocallyUniformlyOn F f p s ↔ ∀ x ∈ s, TendstoUniformlyOnFilter F f p (𝓝[s] x) := by |
simp only [TendstoUniformlyOnFilter, eventually_prod_iff]
constructor
· rintro h x hx u hu
obtain ⟨s, hs1, hs2⟩ := h u hu x hx
exact ⟨_, hs2, _, eventually_of_mem hs1 fun x => id, fun hi y hy => hi y hy⟩
· rintro h u hu x hx
obtain ⟨pa, hpa, pb, hpb, h⟩ := h x hx u hu
exact ⟨pb, hpb, eventually... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Logic.Equiv.Defs
#align_import data.erased from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
/-!
# A type for VM-erased ... | Mathlib/Data/Erased.lean | 131 | 131 | theorem map_out {α β} {f : α → β} (a : Erased α) : (a.map f).out = f a.out := by | simp [map]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 639 | 642 | theorem subset_support_append_right {V : Type u} {G : SimpleGraph V} {u v w : V}
(p : G.Walk u v) (q : G.Walk v w) : q.support ⊆ (p.append q).support := by |
intro h
simp (config := { contextual := true }) only [mem_support_append_iff, or_true_iff, imp_true_iff]
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-c... | Mathlib/ModelTheory/Definability.lean | 167 | 172 | theorem Definable.preimage_comp (f : α → β) {s : Set (α → M)} (h : A.Definable L s) :
A.Definable L ((fun g : β → M => g ∘ f) ⁻¹' s) := by |
obtain ⟨φ, rfl⟩ := h
refine ⟨φ.relabel f, ?_⟩
ext
simp only [Set.preimage_setOf_eq, mem_setOf_eq, Formula.realize_relabel]
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Satisfiability
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import model_theory.graph from "leanprover-community/mathlib"@"e56b8f... | Mathlib/ModelTheory/Graph.lean | 114 | 130 | theorem structure_simpleGraphOfStructure [S : Language.graph.Structure V] [V ⊨ Theory.simpleGraph] :
(simpleGraphOfStructure V).structure = S := by |
ext
case funMap n f xs =>
exact (IsRelational.empty_functions n).elim f
case RelMap n r xs =>
rw [iff_eq_eq]
cases' n with n
· exact r.elim
· cases' n with n
· exact r.elim
· cases' n with n
· cases r
change RelMap adj ![xs 0, xs 1] = _
refine congr rfl... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 1,449 | 1,457 | theorem surjOn_iff_exists_bijOn_subset : SurjOn f s t ↔ ∃ s' ⊆ s, BijOn f s' t := by |
constructor
· rcases eq_empty_or_nonempty t with (rfl | ht)
· exact fun _ => ⟨∅, empty_subset _, bijOn_empty f⟩
· intro h
haveI : Nonempty α := ⟨Classical.choose (h.comap_nonempty ht)⟩
exact ⟨_, h.mapsTo_invFunOn.image_subset, h.bijOn_subset⟩
· rintro ⟨s', hs', hfs'⟩
exact hfs'.surjOn.mon... |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-communi... | Mathlib/Algebra/DirectSum/Decomposition.lean | 127 | 128 | theorem decompose_coe {i : ι} (x : ℳ i) : decompose ℳ (x : M) = DirectSum.of _ i x := by |
rw [← decompose_symm_of _, Equiv.apply_symm_apply]
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Martingale.OptionalStopping
import Mathlib.Probability.Martingale.Centering
#align_import prob... | Mathlib/Probability/Martingale/BorelCantelli.lean | 270 | 274 | theorem Martingale.ae_not_tendsto_atTop_atBot [IsFiniteMeasure μ] (hf : Martingale f ℱ μ)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, ¬Tendsto (fun n => f n ω) atTop atBot := by |
filter_upwards [hf.bddAbove_range_iff_bddBelow_range hbdd] with ω hω htop using
unbounded_of_tendsto_atBot htop (hω.1 <| bddAbove_range_of_tendsto_atTop_atBot htop)
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 574 | 575 | theorem card_support_eq_zero {f : Perm α} : f.support.card = 0 ↔ f = 1 := by |
rw [Finset.card_eq_zero, support_eq_empty_iff]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 142 | 144 | theorem isClosed_iff {s : Set Γ₀} : IsClosed s ↔ (0 : Γ₀) ∈ s ∨ ∃ γ, γ ≠ 0 ∧ s ⊆ Ici γ := by |
simp only [← isOpen_compl_iff, isOpen_iff, mem_compl_iff, not_not, ← compl_Ici,
compl_subset_compl]
|
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Bilin... | Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean | 65 | 83 | theorem IsIntegralClosure.isLocalization [Algebra.IsAlgebraic K L] :
IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L := by |
haveI : IsDomain C :=
(IsIntegralClosure.equiv A C L (integralClosure A L)).toMulEquiv.isDomain (integralClosure A L)
haveI : NoZeroSMulDivisors A L := NoZeroSMulDivisors.trans A K L
haveI : NoZeroSMulDivisors A C := IsIntegralClosure.noZeroSMulDivisors A L
refine ⟨?_, fun z => ?_, fun {x y} h => ⟨1, ?_⟩⟩
... |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Riccardo Brasca
-/
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.Quot... | Mathlib/Analysis/Normed/Group/Quotient.lean | 187 | 190 | theorem QuotientAddGroup.norm_lt_iff {S : AddSubgroup M} {x : M ⧸ S} {r : ℝ} :
‖x‖ < r ↔ ∃ m : M, ↑m = x ∧ ‖m‖ < r := by |
rw [isGLB_lt_iff (isGLB_quotient_norm _), exists_mem_image]
rfl
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/math... | Mathlib/CategoryTheory/Iso.lean | 248 | 250 | theorem hom_eq_inv (α : X ≅ Y) (β : Y ≅ X) : α.hom = β.inv ↔ β.hom = α.inv := by |
erw [inv_eq_inv α.symm β, eq_comm]
rfl
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Topology.Algebra.Ring.Ideal
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.normed_space.units from "leanprover-community/mathl... | Mathlib/Analysis/NormedSpace/Units.lean | 256 | 258 | theorem closure_ne_top (I : Ideal R) (hI : I ≠ ⊤) : I.closure ≠ ⊤ := by |
have h := closure_minimal (coe_subset_nonunits hI) nonunits.isClosed
simpa only [I.closure.eq_top_iff_one, Ne] using mt (@h 1) one_not_mem_nonunits
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Data.Finset.Pointwise
import Mathlib.Data.Finsupp.Indicator
import Mathlib.Data.Fintype.BigOperators
#align_im... | Mathlib/Data/Finset/Finsupp.lean | 101 | 103 | theorem card_pi (f : ι →₀ Finset α) : f.pi.card = f.prod fun i => (f i).card := by |
rw [pi, card_finsupp]
exact Finset.prod_congr rfl fun i _ => by simp only [Pi.natCast_apply, Nat.cast_id]
|
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.SetLike.Fintype
import Mathlib.Algebra.Divisibility.Prod
import Mathlib.RingTheory.Nakayama
import Mathlib.RingTheory.SimpleModule
import Mathlib.Tact... | Mathlib/RingTheory/Artinian.lean | 396 | 411 | theorem isArtinian_of_fg_of_artinian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [IsArtinianRing R] (hN : N.FG) : IsArtinian R N := by |
let ⟨s, hs⟩ := hN
haveI := Classical.decEq M
haveI := Classical.decEq R
have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
refine @isArtinian_of_surjective _ ((↑s : Set M) →₀ R) N _ _ _ _ _ ?_ ?_ isArtinian_finsupp
· exact Finsupp.total (↑s : Set M) N R (fun i => ⟨i, hs ▸ subset_span i.2⟩)
... |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.Topology.QuasiSeparated
#align_import algebraic_geometry.morphisms.quasi_separated from "leanprover-... | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 300 | 307 | theorem exists_eq_pow_mul_of_isAffineOpen (X : Scheme) (U : Opens X.carrier) (hU : IsAffineOpen U)
(f : X.presheaf.obj (op U)) (x : X.presheaf.obj (op <| X.basicOpen f)) :
∃ (n : ℕ) (y : X.presheaf.obj (op U)), y |_ X.basicOpen f = (f |_ X.basicOpen f) ^ n * x := by |
have := (hU.isLocalization_basicOpen f).2
obtain ⟨⟨y, _, n, rfl⟩, d⟩ := this x
use n, y
delta TopCat.Presheaf.restrictOpen TopCat.Presheaf.restrict
simpa [mul_comm x] using d.symm
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 560 | 561 | theorem eval_apply_eq_zero_iff (v : V) : (eval K V) v = 0 ↔ v = 0 := by |
simpa only using SetLike.ext_iff.mp (eval_ker K V) v
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0... | Mathlib/Geometry/Euclidean/MongePoint.lean | 521 | 545 | theorem dist_orthocenter_reflection_circumcenter (t : Triangle ℝ P) {i₁ i₂ : Fin 3} (h : i₁ ≠ i₂) :
dist t.orthocenter (reflection (affineSpan ℝ (t.points '' {i₁, i₂})) t.circumcenter) =
t.circumradius := by |
rw [← mul_self_inj_of_nonneg dist_nonneg t.circumradius_nonneg,
t.reflection_circumcenter_eq_affineCombination_of_pointsWithCircumcenter h,
t.orthocenter_eq_mongePoint, mongePoint_eq_affineCombination_of_pointsWithCircumcenter,
dist_affineCombination t.pointsWithCircumcenter (sum_mongePointWeightsWithCir... |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn, Mario Carneiro
-/
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f :... | .lake/packages/batteries/Batteries/Logic.lean | 142 | 143 | theorem congr_eqRec {β : α → Sort _} (f : (x : α) → β x → γ) (h : x = x') (y : β x) :
f x' (Eq.rec y h) = f x y := by | cases h; rfl
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Da... | Mathlib/Algebra/Group/Subsemigroup/Basic.lean | 283 | 286 | theorem subsingleton_of_subsingleton [Subsingleton (Subsemigroup M)] : Subsingleton M := by |
constructor; intro x y
have : ∀ a : M, a ∈ (⊥ : Subsemigroup M) := by simp [Subsingleton.elim (⊥ : Subsemigroup M) ⊤]
exact absurd (this x) not_mem_bot
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 916 | 927 | theorem LinearIndependent.repr_eq {l : ι →₀ R} {x : span R (range v)}
(eq : Finsupp.total ι M R v l = ↑x) : hv.repr x = l := by |
have :
↑((LinearIndependent.totalEquiv hv : (ι →₀ R) →ₗ[R] span R (range v)) l) =
Finsupp.total ι M R v l :=
rfl
have : (LinearIndependent.totalEquiv hv : (ι →₀ R) →ₗ[R] span R (range v)) l = x := by
rw [eq] at this
exact Subtype.ext_iff.2 this
rw [← LinearEquiv.symm_apply_apply hv.totalEqu... |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprove... | Mathlib/GroupTheory/Perm/List.lean | 156 | 158 | theorem formPerm_apply_get_length (x : α) (xs : List α) :
formPerm (x :: xs) ((x :: xs).get (Fin.mk xs.length (by simp))) = x := by |
rw [get_cons_length, formPerm_apply_getLast]; rfl;
|
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.Algebra.DirectSum.Bas... | Mathlib/RingTheory/AdicCompletion/Functoriality.lean | 169 | 172 | theorem map_comp (f : M →ₗ[R] N) (g : N →ₗ[R] P) :
map I g ∘ₗ map I f = map I (g ∘ₗ f) := by |
ext
simp
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | Mathlib/MeasureTheory/Measure/Portmanteau.lean | 172 | 183 | theorem limsup_measure_closed_le_iff_liminf_measure_open_ge {ι : Type*} {L : Filter ι}
{μ : Measure Ω} {μs : ι → Measure Ω} [IsProbabilityMeasure μ]
[∀ i, IsProbabilityMeasure (μs i)] :
(∀ F, IsClosed F → (L.limsup fun i => μs i F) ≤ μ F) ↔
∀ G, IsOpen G → μ G ≤ L.liminf fun i => μs i G := by |
constructor
· intro h G G_open
exact le_measure_liminf_of_limsup_measure_compl_le
G_open.measurableSet (h Gᶜ (isClosed_compl_iff.mpr G_open))
· intro h F F_closed
exact limsup_measure_le_of_le_liminf_measure_compl
F_closed.measurableSet (h Fᶜ (isOpen_compl_iff.mpr F_closed))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.CategoryTheory.Iso
import Mathlib.RingTheory.Localization.Aw... | Mathlib/RingTheory/RingHomProperties.lean | 132 | 163 | theorem StableUnderBaseChange.mk (h₁ : RespectsIso @P)
(h₂ :
∀ ⦃R S T⦄ [CommRing R] [CommRing S] [CommRing T],
∀ [Algebra R S] [Algebra R T],
P (algebraMap R T) →
P (Algebra.TensorProduct.includeLeftRingHom : S →+* TensorProduct R S T)) :
StableUnderBaseChange @P := by |
introv R h H
let e := h.symm.1.equiv
let f' :=
Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S')
(IsScalarTower.toAlgHom R S S')
have : ∀ x, e x = f' x := by
intro x
change e.toLinearMap.restrictScalars R x = f'.toLinearMap x
congr 1
apply TensorProduct.ext'
intro ... |
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.Algebra.Group.Fin
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.circulant from "leanprover-community/mathlib"@"3e068... | Mathlib/LinearAlgebra/Matrix/Circulant.lean | 166 | 169 | theorem circulant_single_one (α n) [Zero α] [One α] [DecidableEq n] [AddGroup n] :
circulant (Pi.single 0 1 : n → α) = (1 : Matrix n n α) := by |
ext i j
simp [one_apply, Pi.single_apply, sub_eq_zero]
|
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.EisensteinCriterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.eisenstein.basic from "leanprove... | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 128 | 138 | theorem pow_natDegree_le_of_root_of_monic_mem {x : R} (hroot : IsRoot f x) (hmo : f.Monic) :
∀ i, f.natDegree ≤ i → x ^ i ∈ 𝓟 := by |
intro i hi
obtain ⟨k, hk⟩ := exists_add_of_le hi
rw [hk, pow_add]
suffices x ^ f.natDegree ∈ 𝓟 by exact mul_mem_right (x ^ k) 𝓟 this
rw [IsRoot.def, eval_eq_sum_range, Finset.range_add_one,
Finset.sum_insert Finset.not_mem_range_self, Finset.sum_range, hmo.coeff_natDegree, one_mul] at
*
rw [eq_ne... |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.LinearAlgebra.Span
import Mathlib.LinearAlgebra.BilinearMap
#align_import algebra.module.submodule.bilinear from "leanprover-community/mathlib"@"60... | Mathlib/Algebra/Module/Submodule/Bilinear.lean | 129 | 132 | theorem image2_subset_map₂ (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q : Submodule R N) :
Set.image2 (fun m n => f m n) (↑p : Set M) (↑q : Set N) ⊆ (↑(map₂ f p q) : Set P) := by |
rintro _ ⟨i, hi, j, hj, rfl⟩
exact apply_mem_map₂ _ hi hj
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib... | Mathlib/Analysis/Calculus/ParametricIntegral.lean | 75 | 155 | theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜] E} (ε_pos : 0 < ε)
(hF_meas : ∀ x ∈ ball x₀ ε, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
(hF'_meas : AEStronglyMeasurable F' μ)
(h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖)
(bound_... |
have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
have nneg : ∀ x, 0 ≤ ‖x - x₀‖⁻¹ := fun x ↦ inv_nonneg.mpr (norm_nonneg _)
set b : α → ℝ := fun a ↦ |bound a|
have b_int : Integrable b μ := bound_integrable.norm
have b_nonneg : ∀ a, 0 ≤ b a := fun a ↦ abs_nonneg _
replace h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.RowCol
#align_import linear_algebra.matrix.trace from "leanprov... | Mathlib/LinearAlgebra/Matrix/Trace.lean | 183 | 186 | theorem trace_col_mul_row [NonUnitalNonAssocSemiring R] (a b : n → R) :
trace (col a * row b) = dotProduct a b := by |
apply Finset.sum_congr rfl
simp [mul_apply]
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.GCongr
#align_import analysis.convex.function fr... | Mathlib/Analysis/Convex/Function.lean | 860 | 871 | theorem neg_strictConvexOn_iff : StrictConvexOn 𝕜 s (-f) ↔ StrictConcaveOn 𝕜 s f := by |
constructor
· rintro ⟨hconv, h⟩
refine ⟨hconv, fun x hx y hy hxy a b ha hb hab => ?_⟩
simp only [ne_eq, Pi.neg_apply, smul_neg, lt_add_neg_iff_add_lt, add_comm,
add_neg_lt_iff_lt_add] at h
exact h hx hy hxy ha hb hab
· rintro ⟨hconv, h⟩
refine ⟨hconv, fun x hx y hy hxy a b ha hb hab => ?_⟩
... |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algeb... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 415 | 424 | theorem vecMul_injective_iff_isUnit {A : Matrix m m K} :
Function.Injective A.vecMul ↔ IsUnit A := by |
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· rw [← vecMul_surjective_iff_isUnit]
exact LinearMap.surjective_of_injective (f := A.vecMulLinear) h
change Function.Injective A.vecMulLinear
rw [← LinearMap.ker_eq_bot, LinearMap.ker_eq_bot']
intro c hc
replace h := h.invertible
simpa using congr_arg A⁻¹.vecMulLinear... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 1,661 | 1,662 | theorem swap_apply_of_ne_of_ne {a b x : α} : x ≠ a → x ≠ b → swap a b x = x := by |
simp (config := { contextual := true }) [swap_apply_def]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import ... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 663 | 667 | theorem decomp (f : V1 →ᵃ[k] V2) : (f : V1 → V2) = ⇑f.linear + fun _ => f 0 := by |
ext x
calc
f x = f.linear x +ᵥ f 0 := by rw [← f.map_vadd, vadd_eq_add, add_zero]
_ = (f.linear + fun _ : V1 => f 0) x := rfl
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCat... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 761 | 762 | theorem augment_hom_zero (X : CosimplicialObject C) (X₀ : C) (f : X₀ ⟶ X.obj [0]) (w) :
(X.augment X₀ f w).hom.app [0] = f := by | simp
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,098 | 1,100 | theorem inner_orthogonalProjection_left_eq_right [HasOrthogonalProjection K] (u v : E) :
⟪↑(orthogonalProjection K u), v⟫ = ⟪u, orthogonalProjection K v⟫ := by |
rw [← inner_orthogonalProjection_eq_of_mem_left, inner_orthogonalProjection_eq_of_mem_right]
|
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 325 | 337 | theorem memℒp_of_finite_measure_preimage (p : ℝ≥0∞) {f : α →ₛ E}
(hf : ∀ y, y ≠ 0 → μ (f ⁻¹' {y}) < ∞) : Memℒp f p μ := by |
by_cases hp0 : p = 0
· rw [hp0, memℒp_zero_iff_aestronglyMeasurable]; exact f.aestronglyMeasurable
by_cases hp_top : p = ∞
· rw [hp_top]; exact memℒp_top f μ
refine ⟨f.aestronglyMeasurable, ?_⟩
rw [snorm_eq_snorm' hp0 hp_top, f.snorm'_eq]
refine ENNReal.rpow_lt_top_of_nonneg (by simp) (ENNReal.sum_lt_top... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 1,944 | 1,949 | theorem _root_.dfinsupp_sumAddHom_mem [∀ i, AddZeroClass (β i)] [AddCommMonoid γ] {S : Type*}
[SetLike S γ] [AddSubmonoidClass S γ] (s : S) (f : Π₀ i, β i) (g : ∀ i, β i →+ γ)
(h : ∀ c, f c ≠ 0 → g c (f c) ∈ s) : DFinsupp.sumAddHom g f ∈ s := by |
classical
rw [DFinsupp.sumAddHom_apply]
exact dfinsupp_sum_mem s f (g ·) h
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Hom.Defs
#align_import data.matrix.dmatrix from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# Depende... | Mathlib/Data/Matrix/DMatrix.lean | 138 | 140 | theorem map_zero [∀ i j, Zero (α i j)] {β : m → n → Type w} [∀ i j, Zero (β i j)]
{f : ∀ ⦃i j⦄, α i j → β i j} (h : ∀ i j, f (0 : α i j) = 0) :
(0 : DMatrix m n α).map f = 0 := by | ext; simp [h]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 641 | 654 | theorem mem_iInf' {ι} {s : ι → Filter α} {U : Set α} :
(U ∈ ⨅ i, s i) ↔
∃ I : Set ι, I.Finite ∧ ∃ V : ι → Set α, (∀ i, V i ∈ s i) ∧
(∀ i ∉ I, V i = univ) ∧ (U = ⋂ i ∈ I, V i) ∧ U = ⋂ i, V i := by |
simp only [mem_iInf, SetCoe.forall', biInter_eq_iInter]
refine ⟨?_, fun ⟨I, If, V, hVs, _, hVU, _⟩ => ⟨I, If, fun i => V i, fun i => hVs i, hVU⟩⟩
rintro ⟨I, If, V, hV, rfl⟩
refine ⟨I, If, fun i => if hi : i ∈ I then V ⟨i, hi⟩ else univ, fun i => ?_, fun i hi => ?_, ?_⟩
· dsimp only
split_ifs
exacts [... |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 1,036 | 1,037 | theorem lift_mul_left (z) : g (f.sec z).2 * f.lift hg z = g (f.sec z).1 := by |
rw [mul_comm, lift_mul_right]
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Or... | Mathlib/Data/Real/NNReal.lean | 937 | 939 | theorem mul_le_iff_le_inv {a b r : ℝ≥0} (hr : r ≠ 0) : r * a ≤ b ↔ a ≤ r⁻¹ * b := by |
have : 0 < r := lt_of_le_of_ne (zero_le r) hr.symm
rw [← mul_le_mul_left (inv_pos.mpr this), ← mul_assoc, inv_mul_cancel hr, one_mul]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Group.Units
import Ma... | Mathlib/Algebra/Order/Ring/Defs.lean | 360 | 366 | theorem mul_le_mul_of_nonpos_right (h : b ≤ a) (hc : c ≤ 0) : a * c ≤ b * c := by |
obtain ⟨d, hcd⟩ := exists_add_of_le hc
refine le_of_add_le_add_right (a := b * d + a * d) ?_
calc
_ = b * d := by rw [add_left_comm, ← mul_add, ← hcd, mul_zero, add_zero]
_ ≤ a * d := mul_le_mul_of_nonneg_right h <| hcd.trans_le <| add_le_of_nonpos_left hc
_ = _ := by rw [← add_assoc, ← mul_add, ← hc... |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
/-!
# Birkhoff average
In this file we define `birkhoffAverage f g n x` to be
$$
\fr... | Mathlib/Dynamics/BirkhoffSum/Average.lean | 82 | 86 | theorem birkhoffAverage_apply_sub_birkhoffAverage {α M : Type*} (R : Type*) [DivisionRing R]
[AddCommGroup M] [Module R M] (f : α → α) (g : α → M) (n : ℕ) (x : α) :
birkhoffAverage R f g n (f x) - birkhoffAverage R f g n x =
(n : R)⁻¹ • (g (f^[n] x) - g x) := by |
simp only [birkhoffAverage, birkhoffSum_apply_sub_birkhoffSum, ← smul_sub]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mat... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 254 | 269 | theorem finprod_heightOneSpectrum_factorization_principal {I : FractionalIdeal R⁰ K} (hI : I ≠ 0)
(k : K) (hk : I = spanSingleton R⁰ k) :
∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {choose
(mk'_surjective R⁰ k)} : ... |
set n : R := choose (mk'_surjective R⁰ k)
set d : ↥R⁰ := choose (choose_spec (mk'_surjective R⁰ k))
have hnd : mk' K n d = k := choose_spec (choose_spec (mk'_surjective R⁰ k))
have hn0 : n ≠ 0 := by
by_contra h
rw [← hnd, h, IsFractionRing.mk'_eq_div, _root_.map_zero,
zero_div, spanSingleton_zero... |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Ba... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 468 | 480 | theorem isRegular_of_isLeftRegular_det {A : Matrix n n α} (hA : IsLeftRegular A.det) :
IsRegular A := by |
constructor
· intro B C h
refine hA.matrix ?_
simp only at h ⊢
rw [← Matrix.one_mul B, ← Matrix.one_mul C, ← Matrix.smul_mul, ← Matrix.smul_mul, ←
adjugate_mul, Matrix.mul_assoc, Matrix.mul_assoc, h]
· intro B C (h : B * A = C * A)
refine hA.matrix ?_
simp only
rw [← Matrix.mul_one ... |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Noetherian
#align_... | Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 74 | 78 | theorem not_isAssociatedPrime_of_subsingleton [Subsingleton M] : ¬IsAssociatedPrime I M := by |
rintro ⟨hI, x, hx⟩
apply hI.ne_top
rwa [Subsingleton.elim x 0, Submodule.span_singleton_eq_bot.mpr rfl, Submodule.annihilator_bot]
at hx
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 1,954 | 1,961 | theorem injective_foldl_comp {l : List (α → α)} {f : α → α}
(hl : ∀ f ∈ l, Function.Injective f) (hf : Function.Injective f) :
Function.Injective (@List.foldl (α → α) (α → α) Function.comp f l) := by |
induction' l with lh lt l_ih generalizing f
· exact hf
· apply l_ih fun _ h => hl _ (List.mem_cons_of_mem _ h)
apply Function.Injective.comp hf
apply hl _ (List.mem_cons_self _ _)
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.centering from "leanprover-community/mathlib"@"bea6c853b6edbd15e9d0941825abd04d77933ed0"... | Mathlib/Probability/Martingale/Centering.lean | 167 | 171 | theorem predictablePart_bdd_difference {R : ℝ≥0} {f : ℕ → Ω → ℝ} (ℱ : Filtration ℕ m0)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, ∀ i, |predictablePart f ℱ μ (i + 1) ω - predictablePart f ℱ μ i ω| ≤ R := by |
simp_rw [predictablePart, Finset.sum_apply, Finset.sum_range_succ_sub_sum]
exact ae_all_iff.2 fun i => ae_bdd_condexp_of_ae_bdd <| ae_all_iff.1 hbdd i
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.or... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 198 | 200 | theorem neg_rotation_neg_pi_div_two (x : V) :
-o.rotation (-π / 2 : ℝ) x = o.rotation (π / 2 : ℝ) x := by |
rw [neg_rotation, ← Real.Angle.coe_add, neg_div, ← sub_eq_add_neg, sub_half]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 872 | 878 | theorem isPreconnected_setOf_wSameSide (s : AffineSubspace ℝ P) (x : P) :
IsPreconnected { y | s.WSameSide x y } := by |
rcases Set.eq_empty_or_nonempty (s : Set P) with (h | h)
· rw [coe_eq_bot_iff] at h
simp only [h, not_wSameSide_bot]
exact isPreconnected_empty
· exact (isConnected_setOf_wSameSide x h).isPreconnected
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 223 | 228 | theorem IsLindelof.inter_iInter_nonempty {ι : Type v} (hs : IsLindelof s) (t : ι → Set X)
(htc : ∀ i, IsClosed (t i)) (hst : ∀ u : Set ι, u.Countable ∧ (s ∩ ⋂ i ∈ u, t i).Nonempty) :
(s ∩ ⋂ i, t i).Nonempty := by |
contrapose! hst
rcases hs.elim_countable_subfamily_closed t htc hst with ⟨u, ⟨_, husub⟩⟩
exact ⟨u, fun _ ↦ husub⟩
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable
#align_import measure_theory.function.cond... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean | 130 | 169 | theorem ae_eq_of_forall_setIntegral_eq_of_sigmaFinite' (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hfg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ ... |
rw [← ae_eq_trim_iff_of_aeStronglyMeasurable' hm hfm hgm]
have hf_mk_int_finite :
∀ s, MeasurableSet[m] s → μ.trim hm s < ∞ → @IntegrableOn _ _ m _ (hfm.mk f) s (μ.trim hm) := by
intro s hs hμs
rw [trim_measurableSet_eq hm hs] at hμs
-- Porting note: `rw [IntegrableOn]` fails with
-- synthesize... |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 799 | 801 | theorem mk'_eq_iff_eq' {x₁ x₂} {y₁ y₂ : S} :
f.mk' x₁ y₁ = f.mk' x₂ y₂ ↔ f.toMap (x₁ * y₂) = f.toMap (x₂ * y₁) := by |
simp only [f.mk'_eq_iff_eq, mul_comm]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-... | Mathlib/Data/TypeVec.lean | 706 | 709 | theorem dropFun_of_subtype {α} (p : α ⟹ «repeat» (n + 1) Prop) :
dropFun (ofSubtype p) = ofSubtype _ := by |
ext i : 2
induction i <;> simp [dropFun, *] <;> rfl
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated... | Mathlib/MeasureTheory/Measure/Dirac.lean | 45 | 49 | theorem dirac_apply_of_mem {a : α} (h : a ∈ s) : dirac a s = 1 := by |
have : ∀ t : Set α, a ∈ t → t.indicator (1 : α → ℝ≥0∞) a = 1 := fun t ht => indicator_of_mem ht 1
refine le_antisymm (this univ trivial ▸ ?_) (this s h ▸ le_dirac_apply)
rw [← dirac_apply' a MeasurableSet.univ]
exact measure_mono (subset_univ s)
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 577 | 579 | theorem continuousAt_prod_of_discrete_right [DiscreteTopology β] {f : α × β → γ} {x : α × β} :
ContinuousAt f x ↔ ContinuousAt (f ⟨·, x.2⟩) x.1 := by |
simp_rw [← continuousWithinAt_univ]; exact continuousWithinAt_prod_of_discrete_right
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury G. Kudryashov, Scott Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.BigOperators.Finsupp
impor... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 578 | 588 | theorem mul_single_apply_of_not_exists_mul [Mul G] (r : k) {g g' : G} (x : MonoidAlgebra k G)
(h : ¬∃ d, g' = d * g) : (x * single g r) g' = 0 := by |
classical
rw [mul_apply, Finsupp.sum_comm, Finsupp.sum_single_index]
swap
· simp_rw [Finsupp.sum, mul_zero, ite_self, Finset.sum_const_zero]
· apply Finset.sum_eq_zero
simp_rw [ite_eq_right_iff]
rintro g'' _hg'' rfl
exfalso
exact h ⟨_, rfl⟩
|
/-
Copyright (c) 2024 Ira Fesefeldt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ira Fesefeldt
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
/-!
# Ordinal Approximants for the Fixed points on complete lattices
This file sets up the ordinal approximation theory o... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 77 | 85 | theorem lfpApprox_monotone : Monotone (lfpApprox f x) := by |
unfold Monotone; intros a b h; unfold lfpApprox
refine sSup_le_sSup ?h
apply sup_le_sup_right
simp only [exists_prop, Set.le_eq_subset, Set.setOf_subset_setOf, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
intros a' h'
use a'
exact ⟨lt_of_lt_of_le h' h, rfl⟩
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 1,773 | 1,775 | theorem tendsto_comp_val_Iic_atBot [SemilatticeInf α] {a : α} {f : α → β} {l : Filter β} :
Tendsto (fun x : Iic a => f x) atBot l ↔ Tendsto f atBot l := by |
rw [← map_val_Iic_atBot a, tendsto_map'_iff, Function.comp_def]
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 621 | 625 | theorem norm_setIntegral_le_of_norm_le_const_ae {C : ℝ} (hs : μ s < ∞)
(hC : ∀ᵐ x ∂μ.restrict s, ‖f x‖ ≤ C) : ‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal := by |
rw [← Measure.restrict_apply_univ] at *
haveI : IsFiniteMeasure (μ.restrict s) := ⟨hs⟩
exact norm_integral_le_of_norm_le_const hC
|
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.CategoryTheory.Category.Basic
import Mathlib.CategoryTheory.Functor.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Tactic.NthRewrite
imp... | Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean | 120 | 124 | theorem congr_reverse_comp {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :
Quot.mk (@Quotient.CompClosure _ _ redStep _ _) (p.reverse ≫ p) =
Quot.mk (@Quotient.CompClosure _ _ redStep _ _) (𝟙 Y) := by |
nth_rw 2 [← Quiver.Path.reverse_reverse p]
apply congr_comp_reverse
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 471 | 474 | theorem isRadical_vanishingIdeal (s : Set (PrimeSpectrum R)) : (vanishingIdeal s).IsRadical := by |
rw [← vanishingIdeal_closure, ← zeroLocus_vanishingIdeal_eq_closure,
vanishingIdeal_zeroLocus_eq_radical]
apply Ideal.radical_isRadical
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 571 | 577 | theorem liftProp_subtype_val {Q : (H → H) → Set H → H → Prop} (hG : LocalInvariantProp G G Q)
(hQ : ∀ y, Q id univ y) (U : Opens M) :
LiftProp Q (Subtype.val : U → M) := by |
intro x
show LiftPropAt Q (id ∘ Subtype.val) x
rw [← hG.liftPropAt_iff_comp_subtype_val]
apply hG.liftProp_id hQ
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Ring.Subring.Basic
import Mathlib.Topology.Algebra.... | Mathlib/Topology/Algebra/Ring/Basic.lean | 210 | 216 | theorem TopologicalRing.of_addGroup_of_nhds_zero [TopologicalAddGroup R]
(hmul : Tendsto (uncurry ((· * ·) : R → R → R)) (𝓝 0 ×ˢ 𝓝 0) <| 𝓝 0)
(hmul_left : ∀ x₀ : R, Tendsto (fun x : R => x₀ * x) (𝓝 0) <| 𝓝 0)
(hmul_right : ∀ x₀ : R, Tendsto (fun x : R => x * x₀) (𝓝 0) <| 𝓝 0) : TopologicalRing R wher... |
refine continuous_of_continuousAt_zero₂ (AddMonoidHom.mul (R := R)) ?_ ?_ ?_ <;>
simpa only [ContinuousAt, mul_zero, zero_mul, nhds_prod_eq, AddMonoidHom.mul_apply]
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 137 | 138 | theorem symmDiff_of_le {a b : α} (h : a ≤ b) : a ∆ b = b \ a := by |
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 812 | 817 | theorem Raised.right {l x₁ x₂ r₁ r₂} (H : Raised (size r₁) (size r₂)) :
Raised (size (@node' α l x₁ r₁)) (size (@node' α l x₂ r₂)) := by |
rw [node', size_node, size_node]; generalize size r₂ = m at H ⊢
rcases H with (rfl | rfl)
· exact Or.inl rfl
· exact Or.inr rfl
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/ma... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 109 | 118 | theorem Basis.le_span'' {ι : Type*} [Fintype ι] (b : Basis ι R M) {w : Set M} [Fintype w]
(s : span R w = ⊤) : Fintype.card ι ≤ Fintype.card w := by |
-- We construct a surjective linear map `(w → R) →ₗ[R] (ι → R)`,
-- by expressing a linear combination in `w` as a linear combination in `ι`.
fapply card_le_of_surjective' R
· exact b.repr.toLinearMap.comp (Finsupp.total w M R (↑))
· apply Surjective.comp (g := b.repr.toLinearMap)
· apply LinearEquiv.sur... |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Data.Fintype.Card
#align_import data.finset.noncomm_prod f... | Mathlib/Data/Finset/NoncommProd.lean | 393 | 398 | theorem noncommProd_erase_mul [DecidableEq α] (s : Finset α) {a : α} (h : a ∈ s) (f : α → β) (comm)
(comm' := fun x hx y hy hxy ↦ comm (s.mem_of_mem_erase hx) (s.mem_of_mem_erase hy) hxy) :
(s.erase a).noncommProd f comm' * f a = s.noncommProd f comm := by |
classical
simpa only [← Multiset.map_erase_of_mem _ _ h] using
Multiset.noncommProd_erase_mul (s.1.map f) (Multiset.mem_map_of_mem f h) _
|
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Nilpotent.lean | 553 | 564 | theorem nilpotencyClass_le_of_surjective {G' : Type*} [Group G'] (f : G →* G')
(hf : Function.Surjective f) [h : IsNilpotent G] :
Group.nilpotencyClass (hG := nilpotent_of_surjective _ hf) ≤ Group.nilpotencyClass G := by |
-- Porting note: Lean needs to be told that predicates are decidable
refine @Nat.find_mono _ _ (Classical.decPred _) (Classical.decPred _) ?_ _ _
intro n hn
rw [eq_top_iff]
calc
⊤ = f.range := symm (f.range_top_of_surjective hf)
_ = Subgroup.map f ⊤ := MonoidHom.range_eq_map _
_ = Subgroup.map f ... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geo... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 614 | 617 | theorem _root_.Sbtw.oangle_eq_add_pi_left
{p₁ p₁' p₂ p₃ : P} (h : Sbtw ℝ p₁ p₂ p₁') (hp₃p₂ : p₃ ≠ p₂) :
∡ p₁ p₂ p₃ = ∡ p₁' p₂ p₃ + π := by |
rw [← h.oangle₁₂₃_eq_pi, oangle_add_swap h.left_ne h.right_ne hp₃p₂]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Order.RelIso.Set
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintyp... | Mathlib/Data/Finset/Sort.lean | 240 | 244 | theorem orderEmbOfFin_eq_orderEmbOfFin_iff {k l : ℕ} {s : Finset α} {i : Fin k} {j : Fin l}
{h : s.card = k} {h' : s.card = l} :
s.orderEmbOfFin h i = s.orderEmbOfFin h' j ↔ (i : ℕ) = (j : ℕ) := by |
substs k l
exact (s.orderEmbOfFin rfl).eq_iff_eq.trans Fin.ext_iff
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 323 | 331 | theorem changeForm_contractLeft (d : Module.Dual R M) (x : CliffordAlgebra Q) :
-- Porting note: original statement
-- changeForm h (d⌋x) = d⌋changeForm h x := by |
changeForm h (contractLeft (Q := Q) d x) = contractLeft (Q := Q') d (changeForm h x) := by
induction' x using CliffordAlgebra.left_induction with r x y hx hy m x hx
· simp only [contractLeft_algebraMap, changeForm_algebraMap, map_zero]
· rw [map_add, map_add, map_add, map_add, hx, hy]
· simp only [contract... |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.bas... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 246 | 249 | theorem deriv_zero_of_not_differentiableAt (h : ¬DifferentiableAt 𝕜 f x) : deriv f x = 0 := by |
unfold deriv
rw [fderiv_zero_of_not_differentiableAt h]
simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dea... | Mathlib/Data/Multiset/Bind.lean | 301 | 302 | theorem product_cons : s ×ˢ (b ::ₘ t) = (s.map fun a => (a, b)) + s ×ˢ t := by |
simp [SProd.sprod, product]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology... | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 546 | 562 | theorem uniformSpace_eq_inf_precomp_of_cover {δ₁ δ₂ : Type*} (φ₁ : δ₁ → α) (φ₂ : δ₂ → α)
(h_cover : range φ₁ ∪ range φ₂ = univ) :
𝒰(α, β, _) =
.comap (ofFun ∘ (· ∘ φ₁) ∘ toFun) 𝒰(δ₁, β, _) ⊓
.comap (ofFun ∘ (· ∘ φ₂) ∘ toFun) 𝒰(δ₂, β, _) := by |
ext : 1
refine le_antisymm (le_inf ?_ ?_) ?_
· exact tendsto_iff_comap.mp UniformFun.precomp_uniformContinuous
· exact tendsto_iff_comap.mp UniformFun.precomp_uniformContinuous
· refine
(UniformFun.hasBasis_uniformity δ₁ β |>.comap _).inf
(UniformFun.hasBasis_uniformity δ₂ β |>.comap _)
|... |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,069 | 1,071 | theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) :
(G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by |
ext <;> simp [and_assoc, not_or]
|
/-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.MvPolynomial.Symmetric
#align_import ring_theory.polynomial.vieta from "leanprover-community/mathlib... | Mathlib/RingTheory/Polynomial/Vieta.lean | 139 | 145 | theorem _root_.Polynomial.coeff_eq_esymm_roots_of_card [IsDomain R] {p : R[X]}
(hroots : Multiset.card p.roots = p.natDegree) {k : ℕ} (h : k ≤ p.natDegree) :
p.coeff k = p.leadingCoeff * (-1) ^ (p.natDegree - k) * p.roots.esymm (p.natDegree - k) := by |
conv_lhs => rw [← C_leadingCoeff_mul_prod_multiset_X_sub_C hroots]
rw [coeff_C_mul, mul_assoc]; congr
have : k ≤ card (roots p) := by rw [hroots]; exact h
convert p.roots.prod_X_sub_C_coeff this using 3 <;> rw [hroots]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.Dimension.DivisionRing
#align_import... | Mathlib/LinearAlgebra/FiniteDimensional.lean | 492 | 495 | theorem finrank_add_le_finrank_add_finrank (s t : Submodule K V) [FiniteDimensional K s]
[FiniteDimensional K t] : finrank K (s ⊔ t : Submodule K V) ≤ finrank K s + finrank K t := by |
rw [← finrank_sup_add_finrank_inf_eq]
exact self_le_add_right _ _
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 777 | 782 | theorem dist_div_sin_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₃ p₂ / Real.Angle.sin (∡ p₃ p₁ p₂) = dist p₁ p₃ := by |
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.sin_coe, dist_comm p₁ p₃,
dist_div_sin_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inl (right_ne_of_oan... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 489 | 490 | theorem ones_blocksFun (n : ℕ) (i : Fin (ones n).length) : (ones n).blocksFun i = 1 := by |
simp only [blocksFun, ones, blocks, i.2, List.get_replicate]
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.